Tomasz Czyżycki

Generalized discrete orbit function transforms of affine Weyl groups

The affine Weyl groups with their corresponding four types of orbit functions are considered. Two independent admissible shifts, which preserve the symmetries of the weight and the dual weight lattices, are classified. Finite subsets of the shifted weight and the shifted dual weight lattices, which serve as a sampling grid and a set of labels of the orbit functions, respectively, are introduced. The complete sets of discretely orthogonal orbit functions over the sampling grids are found and the corresponding discrete Fourier transforms are formulated. The eight standard one-dimensional discrete cosine and sine transforms form special cases of the presented transforms.

The Tresse theorem and differential invariants for the nonlinear Schrödinger equation

In the present paper by using the Tresse theorem we describe a method of construction of all invariants and the differential invariants for a given Lie group, which means invariants containing derivatives of any order. Some important examples from analysis, geometry and physics are presented. In particular, invariants for the nonlinear Schrodinger equation will be investigated.

Generating Functions for Orthogonal Polynomials of A2, C2 and G2

The generating functions of fourteen families of generalized Chebyshev polynomials related to rank two Lie algebras A 2 , C 2 and G 2 are explicitly developed. There exist two classes of the orthogonal polynomials corresponding to the symmetric and antisymmetric orbit functions of each rank two algebra. The Lie algebras G 2 and C 2 admit two additional polynomial collections arising from their hybrid character functions. The admissible shift of the weight lattice permits the construction of a further four shifted polynomial classes of C 2 and directly generalizes formation of the classical univariate Chebyshev polynomials of the third and fourth kinds. Explicit evaluating formulas for each polynomial family are derived and linked to the incomplete exponential Bell polynomials.

Nonpoint Symmetry and Reduction of Nonlinear Evolution and Wave Type Equations

We study the symmetry reduction of nonlinear partial differential equations with two independent variables. We propose new ansatze reducing nonlinear evolution equations to system of ordinary differential equations. The ansatze are constructed by using operators of nonpoint classical and conditional symmetry. Then we find solution to nonlinear heat equation which cannot be obtained in the framework of the classical Lie approach. By using operators of Lie-Backlund symmetries we construct the solutions of nonlinear hyperbolic equations depending on arbitrary smooth function of one variable too.

Symmetry and Solution of Neutron Transport Equations in Nonhomogeneous Media

We propose the group-theoretical approach which enables one to generate solutions of equations of mathematical physics in nonhomogeneous media from solutions of the same problem in a homogeneous medium. The efficiency of this method is illustrated with examples of thermal neutron diffusion problems. Such problems appear in neutron physics and nuclear geophysics. The method is also applicable to nonstationary and nonintegrable in quadratures differential equations.

Group Transformations with Discrete Parameter and Invariant Schrödinger Equation

We propose a method for the construction of potentials and nonlinearities, for which the Schrodinger equation is invariant with respect to a group transformations with discrete parameters. This is a generalization of the known Lie symmetry of differential equations.

Equivalence groups of differential equations and their applications to mathematical physics problems

In this paper the notion of a group of equivalence transformations of the family of differential equations is considered. Moreover invariants of Lie groups are investigated. By using these tools some problems from mathematical physics such as: transformations of potential in the nonlinear Schrodinger equation, neutron diffusion problem and relations in Maxwell and material equations have been studied.